A New Hybrid Automaton Framework with Partial Differential Equation Dynamics

This paper presents the syntax and semantics of a novel type of hybrid automaton (HA) with partial differential equation (PDE) dynamic, partial differential hybrid automata (PDHA). In PDHA, we add a spatial domain $X$ and harness a mathematic conception, partition, to help us formally define the spatial relations. While classically the dynamics of HA are described by ordinary differential equations (ODEs) and differential inclusions, PDHA is capable of describing the behavior of cyber-physical systems (CPS) with continuous dynamics that cannot be modelled using the canonical hybrid systems' framework. For the purposes of analyzing PDHA, we propose another model called the discrete space partial differential hybrid automata (DSPDHA) which handles discrete spatial domains using finite difference methods (FDM) and this simple and intuitive approach reduces the PDHA into HA with ODE systems. We conclude with two illustrative examples in order to exhibit the nature of PDHA and DSPDHA.